Testing the W-exchange mechanism with two-body baryonic B decays

The role of $W$-exchange diagrams in baryonic $B$ decays is poorly understood, and often taken as insignificant. We show that charmful two-body baryonic $B\to {\bf B}_c \bar {\bf B}'$ decays, where ${\bf B}_c$ is an anti-triplet or a sextet charmed baryon, and ${\bf \bar B}'$ an octet charmless (anti-)baryon, provide a good test-bed for the study of the $W$-exchange topology, whose contribution is found to be non-negligible. The calculated branching ratio of the $\bar B^0\to\Lambda^+_c\bar p$ decay, \mbox{$\mathcal{B}(\bar B^0\to\Lambda^+_c\bar p)=(1.6^{+0.9}_{-0.7})\times10^{-5}$}, is in good agreement with experimental data. Its cousin $\bar B_s^0$ mode, $\bar B_s^0\to\Lambda^+_c\bar p$, is a purely $W$-exchange decay, hence is naturally suited for the study of the role of the $W$-exchange topology. We predict ${\cal B}(\bar B_s^0\to\Lambda^+_c\bar p)=(1.2^{+0.7}_{-0.5})\times 10^{-6}$, a relatively large branching ratio to be tested with a future measurement by the LHCb collaboration. Other predictions, such as ${\cal B}(\bar B^0\to\Xi_c^+\bar\Sigma^-)=(1.8^{+1.0}_{-0.8})\times 10^{-5}$, can be tested with future Belle II measurements.


I. INTRODUCTION
Decays of B mesons to multi-body baryonic final states, such as B → BB ′ M and BB ′ MM ′ , where B (M) represents a baryon (meson), have been richly studied. Their branching ratios are typically at the level of 10 −6 [1]. These relatively large branching ratios are due to the fact that the baryon-pair production tends to occur in the threshold region of m BB ′ ≃ m B + mB′, where the threshold effect with a sharply raising peak can enhance the branching ratio [2][3][4][5]. On the other hand, without the recoiled meson(s) to carry away the large energy release, the two-body B → BB ′ decays proceed at the m B scale, several GeV away from the threshold region, resulting in the suppression of their decay rates [6,7].
The tree-level dominated B → BB ′ decays can proceed through the W -exchange, emission and annihilation diagrams, depicted in Figs. 1(a,b,c), respectively. However, the Wexchange (annihilation) process is regarded as helicity suppressed [11,12], and hence neglected in theoretical studies [13][14][15][16][17]. Moreover, one also neglects the penguin-level gluonexchange (annihilation) contributions, which leads to B(B 0 s → pp) ≃ 0 [17]. Consequently, the observation of theB 0 s → pp decay would provide valuable information on whether contributions from the exchange (annihilation) processes play a significant role. The smallness of the current upper bound on its branching ratio, B(B 0 s → pp) < 1.5 × 10 −8 [8], indicates an experimentally difficult decay mode to study the role of W -exchange diagrams.
On the other hand, experimental data show that where B c denotes a charmed baryon [1]. The set of measured B → B cB ′ branching ratios is nevertheless scarce [1,18,19]: With significantly larger decay rates, charmful two-body baryonic B decays offer an interesting and suitable environment in which to study and test the role of the W -exchange (annihilation) mechanism.
As in the case of the charmless final states considered above, proceed through both the W -exchange and W -emission diagrams. Again, theoretical studies regard the W -exchange diagram as helicity-suppressed, and take the W -emission diagram as the dominant contribution [13,14,[20][21][22]. With A(B → ℓν ℓ ) ∝ m ℓū (1 + γ 5 )v, the small m ℓ is responsible for the helicity suppression. On the other hand, the amplitude of , the W -exchange (annihilation) process in B → B cB ′ is clearly helicity allowed, indicating that neglecting its contribution may not be a valid assumption to make.
For completeness, we note that the theoretical studies in Refs. [23][24][25][26] also considered the exchange and annihilation contributions in B → BB ′ and D + s → pn.  , Ω 0 c ), respectively, andB ′ an octet charmless (anti-)baryon. The decays with the decuplet charmless (anti-)baryons are excluded from the calculations in this paper due to the lack of the corresponding timelike baryon form factors.
We show in Table I the amplitudes involved in the interestingB 0 (s) → B cB ′ modes.
The decay rate of modes that can only occur through the W -exchange diagram would be vanishingly small by construction if the importance of these diagrams was to be insignificant: None of these relations has yet been verified experimentally.
The relevant part of the Hamiltonian for theB 0 (s) → B cB ′ decays has the following form [27]: where G F is the Fermi constant, V ij stand for the CKM matrix elements, and (q 1 In the factorization approach, the W -exchange amplitude ofB 0 (s) → B cB ′ is given by [20,26] where q = d(s) forB 0 (s) , and a 2 = c eff 2 + c eff 1 /N c consists of the effective Wilson coefficients (c eff 1 , c eff 2 ) = (1.168, −0.365) and the color number N c . The matrix elements in Eq. (4) are defined as [1,21] (for simplification, B denotes B 0 (s) in the remainder of this section) where f B is the B meson decay constant and q µ = (p Bc + pB′) µ the momentum transfer.
where (|ξ|, |ζ|) are given in Table II, and (C Λ + cp f,g , C Σ + cp f,g ) are taken as the theoretical inputs, which can be calculated in QCD models.
We compute the branching ratios from the decay-rate equation for two-body decays, given where τ B denotes the B meson lifetime.

III. NUMERICAL ANALYSIS
In the numerical analysis, we use the Wolfenstein parameterization for the CKM matrix elements, given by [1] ( where λ = 0.22453 ± 0.00044 and A = 0.836 ± 0.015, together with the B meson decay constants (f B , f Bs ) = (0.19, 0.23) GeV [1]. For the form factors in Eqs. (5,6) and (7,8), we adopt the ones for 0 → Λ + cp , Σ + cp from the QCD light-cone sum rules [28]: where the dominant uncertainties come from the parameters input in QCD, such as the In the generalized edition of the factorization approach [29], N c is Decay modes B × 10 6 Decay modes B × 10 8 3) commonly appears in the interpretation of charmful b-hadron decays [21,[30][31][32]. All predictedB 0 (s) → B cB ′ branching ratios are given in Table III, with n = 1 assigned for the monopole behavior.

IV. DISCUSSIONS AND CONCLUSIONS
One of the theoretical supports to neglect the W -exchange mechanism is in analogy with the B → ℓν ℓ decays [12] , which are helicity suppressed. In fact, we obtain where m 2 µ ≃ 0.01 GeV 2 corresponds to B(B → µν µ ) = (5.3 ± 2.0 ± 0.9) × 10 −7 [33,34]. Nonetheless, with m +,− ≡ m Bc ± mB′ and R m ≡ (m − /m + ) 2 , m 2 + in B → B cB ′ causes no suppression, while m Bc can be as large as (2.3−2.5) GeV. As a consequence, the W -exchange contribution in B → B cB ′ decays is not necessarily negligible.
With the W -exchange contribution alone, the branching ratio ofB 0 → Λ + cp was once calculated to be as small as 4.6 × 10 −7 [35], which has been taken as theoretical support for neglecting the W -exchange diagram. Note that the calculation used N c = 3 in the naive factorization and n = 2 for baryon form factors with dipole behavior, whereas C Λ + cp f,g are not clearly given. However, being sensitive to the color number, a 2 with N c = 3 has been shown to fail to accommodate the non-factorizable effects [29]; besides, while it is poorly understood whether n = 1 or n = 2 is preferred [21], n = 1 is most often used in recent QCD models [28,36,37]. In the generalized factorization, together with n = 1 for the monople-type form factors, we obtain which are consistent with the current data, cf. Eq. (1). Note that B(B 0 → Σ + cp ) is much smaller than B(B 0 → Λ + cp ) due to the small C Σ + cp g in Eq. (11).
Our calculation of B(B 0 → Λ + cp ) ≃ 1.6×10 −5 is as large as the values of (2.3−5.1)×10 −5 and 1.1 × 10 −5 from the perturbative QCD approach and pole model, respectively [20][21][22], which only take into account the W -emission contribution. This causes the confusion that, if the W -exchange and emission contributions are compatible, the combination of the branching ratios should exceed the experimental data. For clarification, we study decays other thanB 0 → Λ + cp , Σ + cp , which receive contributions from both A ex and A em . By taking A ex as the primary contribution, we predict with R Bs ≡ |(V us f Bs )/(V ud f B )| 2 = 0.075. In future measurements, the deviation of R Bs from the theoretical prediction can be used to evaluate the size of A em in the decays.
Being pure W -exchange processes,B 0 → Ξ + which are all within the capability of the current B factories.
In summary, we have studied the charmful two-body baryonic B 0 (s) → B cB ′ decays, with